QUADRATIC FORMULA - projectile motion
1) Milt Famey, the baseball player, hits a pop fly to the infield. It goes upward
with an initial velocity of 97.5 ft/sec. Assume he hits the ball from a height of 3
feet.
a) Write an equation relating his height to the time in seconds (use variables “h”
and “t”).
b) What is his altitude after 2 seconds?
c) When is it 25 feet above where it was hit?
d) When does the ball hit the ground?
e) When is the ball at its highest point?
f) What is the highest point (in feet) of the ball?
2) Wile E. Coyote is standing on a springboard atop a high cliff. Roadrunner drops
a boulder on the other end of the springboard, sending Wile up with an initial
velocity of 44.5 m/sec (assume starting height is “zero”).
a) Write an equation relating the height Wile in relationship to the time in seconds
(use variables “h” and “t”).
b) How high will Wile be after 4 seconds?
c) What is the highest point that Wile reaches and when does he reach that height?
d) When does Wile again reach the level of the springboard?
e) On the way down, Wile misses the cliff! At what time does he land in the river,
360 m below the top of the cliff?
3) Snoopy is flying in his Sopwith Camel. He fires at the Red Baron. The bullet
has an initial upward velocity of 390 ft/sec. Assume the starting height is “zero”.
a) Write an equation relating the height of the bullet in relationship to the time in
seconds (use variables “h” and “t”).
b) The Red Baron is 1300 ft above Snoopy. When will the bullet first reach his
altitude?
c) The bullet misses on the way up. When could it hit the Red Baron on its way
back down?
d) If the bullet also misses the Red Baron on the way down, when will it be back at
the level of Snoopy’s Sopwith Camel?
e) If it misses Snoopy on the way down, when will it hit the ground, 900 ft below
where it was fired?
4) Wimbledon champs Venus and Serena Williams are relaxing with a friendly
tennis game. Venus holds her tennis racket horizontally at a height of 1.5 m above
the ground. She returns an incoming ball at a speed of 10 m/s.
a) Write an equation that relates the height of the ball in relationship to the time in
seconds.
b) If Serena hits the ball just as it hits its maximum height, what is that height and
after how many seconds does she hit it?
c) If Serena runs to the ball and returns it just before it hits the ground, how much
time did she have to get to the ball?
PROJECTILE MOTION – 4 problems homework answers & explanation…
1) v = 97.5, starting height = 3 ft, it is in feet so you use the “-16” equation.
a) h = -16t2 + 97.5t + 3
b) this is t = 2, so you plug it into the equation in “a” and find the answer = 134 seconds
c) this means the ending height is 25 ft (h = 25) so you need to first set up the equation.
25 = -16t2 + 97.5t + 3 and then use the quadratic formula (don’t forget to put into standard form
first – move that 25!). Answer – at both 0.23 and 5.86 seconds
d) now the ending height is 0 – so use the equation 0 = -16t2 + 97.5t + 3 and solve using
the quadratic formula (already in standard form). Answer – after 6.1 seconds
e) Highest point – means find the vertex! Remember the x-value of the vertex is the time
in seconds and the y-value is the height. Use –b/(2a) to find the x-value, find the y-value by
plugging result back into original equation (in part a). answer 3.05 seconds
f) use the y-value of the vertex for height – answer 151.54 seconds
2) v = 44.5, starting height = 0 and this is meters, so you use the “-4.9” equation
a) h = -4.9t2 + 44.5t + 0
b) this means t = 4, plug it into the equation “a” and find the answer = 99.6 m
c) highest point spells VERTEX! he reaches 101.03 m at 4.54 seconds
d) ending height = 0 (back at springboard) so use the equation 0 = -4.9t2 + 44.5t + 0 and
solve using the quadratic formula (already in standard form)Answer = 9.08 seconds
e) now ending height = -300! Use the equation -300 = -4.9t2 + 44.5t + 0 and remember
to put it into standard form, then use the quadratic formula to solve. Answer = 14.24 seconds
3) In this problem v = 390, starting height we’ve been told to use h 0 = 0.
a) h = -16t2 + 390t + 0
b) ending height = 1300, so use the equation 1300 = -16t2 + 390t + 0 - use the quadratic
formula – gotta put into standard form first!! And you will get two positive answers. The smaller
tells you when the bullet first reaches 1300 and the larger tells you when it reaches 1300 ft on
the way down (answer to part c). answer = 3.98 seconds
c) answer = 20.39 seconds (see “b” for explanation)
d) ending height is now 0 (back at snoopy’s height). Use the equation 0 = -16t2 + 390t +
0 and solve using the quadratic formula. Answer = 24.38 seconds
e) ending height is now -900 feet. Use -900 = -16t2 + 390t + 0 and solve using
quadratic formula (put in standard form first!). Answer = 26.5 seconds
4) in this problem, starting height (h0) = 1.5 meters, v = 10.
a) h = -4.9t2 + 10t + 1.5
b) maximum height spells VERTEX! The ball reaches 6.6 m after 1.02 seconds
c) ending height = 0 so use the equation 0 = -4.9t2 + 10t + 1.5 and solve using the
quadratic formula. Answer – she must get to the ball within 2.18 seconds.